Searching for closed essential surfaces in knot complements

Document Type

Presentation Abstract

Presentation Date

2-25-2019

Abstract

Culler-Shalen theory uses a 3-manifold’s (P)SL(2,C) character variety to construct essential surfaces in the manifold. It has been a fundamental tool over the last 35 years in low-dimensional topology. Much of its success is due to a solid understanding of the essential surfaces with boundary that can be constructed with the theory. It turns out, however, that not every surface with boundary is detected. One can also construct closed essential surfaces within this framework. In this talk, we will discuss a module-theoretic perspective on Culler-Shalen theory and apply this perspective to show that there are knot complements in S3 which contain closed essential surfaces, none of which are detected by Culler-Shalen theory. As a corollary, we will construct an infinite family of closed hyperbolic Haken 3-manifolds whose representations into PSL(2, C) have traces which are integral (over Z).

Additional Details

Monday, February 25, 2019 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109

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